I'm way too late for some of this. Rebuttals have bounced back and forth, but I don't want to throw out so much stuff.

Aya stuff
Really, a lot of my beef with Aya boils down to how poorly the fan community has used her. There was a time when everyone was contractually obligated to have her show up in every last ounce of doujin material, regardless of whether she fit or not. Said appearance largely involved intruding into a scene, doing nothing but having a spaz fit while spouting "Ayaya hot scoop," then leaving. I think it's Morino Hon's Yakumo-ke that serves as a great example; a doujin about the Yakumos and Aya gets more screen time than Ran or Yukari. If she's intended to be a main character instead, the doujin will all too frequently involve a race or show of speed by default which she will win without fail. She gets a bit coddled in canon as she's so powerful, but fanon borders on abusive. Her popularity has settled in somewhat and I've seen her around less and less, which has caused my opinion of her to jump to neutral. Doesn't take too much, but for her it was quite a move.
I'd also like to put Remilia out there. See Aya above. Part of it is that her schtick is her charisma and bratty, blue-blooded 10 year olds just don't feel charismatic. Also, regardless of how things go, no one's permitted to be rebellious at the SDM. Everyone loves Ojou-sama unconditionally regardless of circumstances, actions, or character personality. There's a half-dozen doujins out there in which Meiling is portrayed as an awesome force of nature who matures from a raw survival-of-the-fittest monster into a benevolent folk hero for the locals. Without fail, she then gets her teeth kicked in by Remilia for no discernible reason and Meiling's nothing but a clumsy teddy bear with nothing but high praise for Remi after that. It just doesn't mesh with storytelling and feels backwards. The SDM has always been hugely popular, which means stuff with Remi appears a lot. I've seen a few too many doujins in which the status quo is kept at the cost of all the other characters associated with her and it's such a drag.
Byakuren stuff
OH GOODY Paz, Monhan, and the A List cut this down from a 6 lunch hour ordeal to a 3 lunch hour one. Building on what's been stated so far:
Byakuren's EVIL
I swear, after reading all of the material a dozen times over, I'm still puzzled to see what only feels like a deliberately negative interpretation of a character that was made to be the opposite. It's really driven me to take a closer look at the character and the result has been the opposite of people's intentions. As for using/corrupting/manipulating people, not only is Byakuren not inclined to that sort of behavior and doesn't have a reason for doing so, but there's little to show that she'd even be mentally capable of manipulating people in the first place. Byakuren's pretty easy to figure out since she's a heart-on-your-sleeve type. Imperfect? Yup. Naive? Sure. EEEE-VILLL? "No, Byakuren is really a nice person." So sayeth the ZUN. He typically shies away from directly stating a character's personality attributes, leaving it for important and succinct summation of a character while showing instead of telling for everything else. He doesn't futz around at all with Byakuren. A very rare and distinctly positive statement from the man himself which few other characters have received.
As for being flawed, well yeah, but in all of my experiences, there's two schools of thought on the character: Byakuren is a still-very-human individual with lofty ideals that fall short when reality sets in which stops her from being a Mary Sue, or Byakuren is someone who simply pretends to be delusional as she's actually an evil-just-because caricature of a James Bond villain. It always seems to be one or the other, and there's more disconnect between the latter than the former considering what's actually stated regarding the character.
She helps youkai
The idea that Byakuren's philosophy regarding youkai can't work at all comes from a standpoint of a few people whose only means with dealing with youkai of any sort involves extermination and little else, so it wouldn't stick with them in the first place. When all you have is a hammer, a hammer is all you'll use. This is strangely hypocritical as Reimu & Co. hang out with youkai constantly and really haven't exterminated a (main character) youkai since... Anyway, Reimu doesn't even want to resort to such measures if possible and Byakuren provides a means to those ends. If she had zero success with her methods, the Myouren temple would have already went kaput. On the contrary, it's stated that the place is thriving despite some lingering unease between the two different kinds of attendees.
Byakuren provides a means of pacification as opposed to extermination. It's a means of self-realization for any youkai open to the idea. Any youkai that would antagonize humans is banned. She isn't above discipline for any misbehavior, even for stuff as light as Kyouko being disruptively noisy and not just her noisy self. The temple at large has even influenced dangerous youkai like Murasa and that results in making Komachi's (who IS Death and thus not exactly likely to bite it) day a bit wetter than drowning someone else. Byakuren certainly can't keep tabs on everyone, but I've always found it hard to believe she's so permissive of any trouble because there's so much evidence indicating its opposite. Byakuren just letting things slide would also work against her goals as it'll trouble anyone associated with Myouren temple and work against what's established there. Though her teachings might not reach all youkai, the baseline is a net positive. She may not exterminate purely evil youkai, but that's not her style. Forcing such a duty on her crosses a moral line. It's not in the job description because that's someone else's job.
She hurts youkai
Stating that this philosophy is dangerous to youkai just doesn't seem to hold as much water as it could since she's more learned in their needs and behaviors and wouldn't bother with it if it did any damage or even destroyed them before reaching enlightenment. Byakuren confirms that youkai are capable of such an achievement and nobody offers much in the way of a point to the opposite. Delve into it any further and it seems to hit an Occam's Razor. And this is Gensokyo! It exists for creatures like youkai and other oddities and goddeities, but requires the presence of humans. None of them are going to go away anytime soon. Byakuren's ideals and goals fit perfectly here and even at their worst still remain a preferable alternative to the status quo. This is even buffered by spell card rules, discouraging the previous youkai-eat-humans-exterminate-youkai cycle. The rule sticks the two in a room, the Myouren temple gives them a reason not to want to bite each other's heads off.
Vanity
The vanity thing never stuck with me because it only seems to come up in passing and is stated in such a way that it's vain to reverse aging at all. Truly, if anyone would pick an age to stay at, it would be when they were at their prime. It never really felt vain for her to choose her late 20's as opposed to an age at which she'd best be described as losing her grip on her faculties and osteoporitic. Staying in an old form would constantly work against her strengths and you can't finish what you set out to accomplish if you're dead and gone. No other point is ever brought up about any concern for her physical appearance or any interpretation of vanity in any capacity, especially not to the point where one's ego would become involved.
I just want to say she's not perfect
Really I don't think her die-hard fans see her as perfect, but her mindset is benevolent while her faults are things that so many people can relate to. She feels like more of a person than some other characters in the series and still carries the burden of her idealism in an admirable manner, hence her draw.
Miko stuff
Byakuren's inverse, so this becomes low-hanging fruit. Suffice to say, Miko's never getting a truly fair look from me because my only real problem with the human race falls to an extraordinarily thin band of personalities that combine pride, greed, manipulation, self-serving egotism, power, and incompetence. What few people I've ever made enemies with are best described as "basically Miko." She's a disaster in her backstory and learns little if anything from it. I gave her a chance for HM and the selfish might-makes-right attitude made me regret it.
TD is basically an in-joke
Yukari and Mamizou have other similarities, too. They both always seem to have that wry smirk on their face, generally act in their own interest but aren't beyond assisting others when the opportunity arises, and both heavily influence a local human.